In math, the
symbol°
is commonly used to produce a function defined as the composition of two
functions: f ° g y
is defined as f (g y) .Normally, such composed functions are
only defined to apply to a single scalar argument.

J provides compositions effected by five distinct conjunctions,
as well as compositions effected by isolated sequences of verbs:
hooks and forks, and longer trains formed from them.
The five conjunctions are & &. &: @ and @: ,
the conjunctions @ and @: being related in
the same manner as & and &: .

The conjunction & is closest to the
composition°
used in math, being identical to it when used for two scalar (rank zero)
functions to produce a function to be applied to a single scalar argument.
However, it is also extended in two directions:

1.

Applied to one verb and one noun it produces
a monadic function illustrated by the cases 10&^.
(Base ten logarithm) and ^&3 (Cube).

2.

Applied to two verbs it produces (in addition to the
monadic case used in math) a dyadic case defined
by: x f&g y ↔ (g x) f (g y) .
For example, x %&! y is the quotient of
the factorials of x and y .

The conjunction &. applies only to verbs,
and f&.g is equivalent to f&g except that
the inverse of g is applied to the final result. For example:

3 +&^. 4 3 +&.^. 4
2.48491 12

For scalar arguments the functions f&:g
and f&g are equivalent, but for more general
arguments, g applies to each cell as dictated by its ranks.
In the case of f&g,the function f
then applies to each result produced; in the case of f&:g
it applies to the overall result of all of the cells. For example: